132 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



be so bounded that, within it, no branch of go merges into another ; 

 such a branch, therefore, represents a possible mode of fluid motion. 

 The desired object will be attained when the region of oo is appropri- 

 ately bounded. 



In reference to the boundary of the region of oo it is recognized, first, 

 that it is a line that returns into itself and without cutting itself and 

 that consists of parts for which >/• has a constant value and of parts 

 for which <p has an indefinitely large positive or an indefinitely large 

 negative value. 



Within the region of &?,/(&?) is a single-valued function of oo. If we 

 had adopted an expression for f (go) that represented a many-valued 

 function, then at its cusp point should start the sections for which t/> 

 has a constant value. 



Furthermore yf f(oo)f(oo)—l should also be made a single- valued 

 function of oo. in that through those points for which /(&?) = ± 1, the sec- 

 tions pass for which >/• lias a constant value. For any point of the 

 region of oo the sign of the radical quantity is still at our disposal. It' 

 points occur for which /(ca) is infinite or infinitely great,* then for one 

 of these points we may make 



Vf(G0)f(G0)-l=+f(G0) 



and assume that tbis equation holds good for them all. 



It is further assumed that the function /(co) is only infinite at its cusp 

 points if it is so anywhere, and even here it is infinite only in such a way 

 that if /(coo) is infinite then (go— oo )f(oo) approximates to zero when oo 

 has a value approximating that of oo . 



Within the designated region of oo therefore z is a single-valued 

 function of this variable and such that it is never infinite. 



Now consider oo as a function of z. The region of z that corresponds 



to the adopted region of oo does not extend through infinity, and is 



bounded by a line that returns into itself aud which is made up of the 



lines whose equations are cp=— ao and r^= + co and of stream lines; a 



certain portion of the latter can be considered as a free boundary of 



the moving fluid, the other part cau be considered as a fixed wall. 



Within this region of z, go has no cusp point, since at no point of it 



dz 

 does -y- become zero. Therefore under the condition that the boundary 



of the region of z shall not intersect itself, go becomes within that 

 region a single valued function of z. 



This function of z is completely determined as soon as one has found 

 a single value of z corresponding to a given value of oo. 



(I.) An example that constitutes a generalization of the case treated 

 of by Helmholtz is obtained if we put 



_ /(Gj)=7v + e- ( » 



* By infinite, I designate the reciprocal of zero, but by infinitely great, the recipro- 

 cal of an infinitely small quantity. 



